A new approach for Kirchhoff-Love plates and shells Walter Zulehner Institute of Computational Mathematics JKU Linz, Austria AANMPDE 10 October 2-6, 2017, Paleochora, Crete, Greece Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 1 / 30
Acknowledgment Joint work with Wolfgang Krendl, Katharina Rafetseder The research was funded by the Austrian Science Fund (FWF): S11702-N23 Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 2 / 30
Outline 1 The Kirchhoff plate bending problem 2 Extension to Mindlin-Reissner plates? 3 Extension to Kirchhoff-Love shells 4 Numerical results 5 Conclusion and outlook Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 3 / 30
Kirchhoff plate bending problem Figure: Geometry of a shell Unknown: covariant components of the displacement u = (u i ) Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 4 / 30
Kirchhoff plate bending problem plate: S is flat, for simplicity: θ = Id pure bending: u 3 = w. dimensional reduction of 3D elasticity for Ω = ( t/2, t/2) find w W H 2 () which minimizes J(w) = 1 C 2 w : 2 w dx 2 with Cε = 2µ ( ε + ν ) (tr ε)i 1 ν g w dx Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 5 / 30
Kirchhoff plate bending problem (kinematic) boundary conditions: = γ c γ s γ f if the plate is clamped on γ c simply supported on γ s then free on γ f W = {v H 2 () : v = 0, n v = 0 on γ c, v = 0 on γ s }. variational formulation: find w W such that C 2 w : 2 v dy = g v dx for all v W, Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 6 / 30
Kirchhoff plate bending problem quantities of interest: w vertical deflection M = C 2 w bending moments Mixed variational formulation: Find M V and w Q such that C 1 M : L dy + L : 2 w dy = 0 for all L V, M : 2 v dy = g v dx for all v Q, e.g., with the function spaces V = L 2 (Ω) sym, Q = W. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 7 / 30
Kirchhoff plate bending problem An different choice for Q with less smoothness: if W = {v H 2 () : v = 0, n v = 0 on γ c, v = 0 on γ s }, then Q = {v H 1 () : v = 0, on γ c, v = 0 on γ s } = H 1 0,γ c γ s (). The choice for V : transposition method b(v, L) = L : 2 v dy Bv, L B L, v, where B L, v is well-defined for v Q. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 8 / 30
Kirchhoff plate bending problem The linear operator B: B : D(B) Q Y, here with domain D(B) = W and Y = L 2 (Ω) sym, is a densely defined linear operator and unbounded operator in Q. Its adjoint operator is given by the identity B : D(B ) Y Q and all L from the domain B L, v = Bv, L for all v W D(B ) = {L Y : Bv, L c v Q for all v D(B)} Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 9 / 30
Kirchhoff plate bending problem standard integration by parts: L : 2 v dy = Div L v dy + = Div L v dy + (Ln) v ds L nt t v ds + L nn n v ds?? L nn = 0 on γ s γ f (L nn = Ln n, L nt = Ln t) = γ s : D(B ) = {L L 2 (Ω) sym : div Div L H 1 ()} Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 10 / 30
Kirchhoff plate bending problem Lemma Let L L 2 (Ω) sym C 1 (). Then L D(B ) if and only if L nn = 0 on γ s γ f and L nt x = 0 on interior corner points x of γ f. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 11 / 30
Kirchhoff plate bending problem Notations for B = 2 = Grad grad: with B = div Div, D(B ) = H(div Div, ; Q ) sym. H(div Div, ; Q ) sym = {L L 2 (Ω) sym : L : 2 v dy c v H 1 () for all v W } Observe H 1 0 () sym H(div Div, ; Q ) sym L 2 (Ω) sym Bernardi/Girault/Maday (1992), Z. (2015), Pechstein/Schöberl (2011), Rafetseder/Z. (2017) Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 12 / 30
Kirchhoff plate bending problem Mixed variational formulation of Find M V and w Q such that C 1 M : L dy + div Div L, w = 0 for all L V, div Div M, v = g v dx for all v Q, with the function spaces V = H(div Div, ; Q ) sym, Q = H 1 0,γ c γ s (), equipped with norms v Q = v 1 and L V = L div Div;Q, given by L 2 div Div;Q = L 2 L 2 (Ω) + div Div L 2 Q. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 13 / 30
Kirchhoff plate bending problem Theorem The mixed problem is well-posed in V Q = H(div Div, ; Q ) sym H 1 0,γ c γ s (). Corollary The primal variational problem and the mixed problem are equivalent: If w W solves the primal problem, then M = C 2 w V and (M, w) solves the mixed problem. If (M, w) V Q solves the mixed problem, then w W and w solves the primal problem. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 14 / 30
Kirchhoff plate bending problem Theorem Let be simply connected. M H(div Div, ; Q ) sym if and only if M = p I + symcurl φ with with Ψ p = p Q = H 1 0,γ c γ s () and φ Ψ p { ψ (H 1 ()) 2 : t ψ, v Γ = p n v ds Γ } for all v W ( ) ( 1 φ1 φ =, sym Curl φ = 2 φ 1 2 ( ) 2φ 2 1 φ 1 ) φ 1 2 2 ( 2φ 2 1 φ 1 ) 1 φ 2 Krendl/Rafetseder/Z. (2014,2016), Rafetseder/Z. (2017) Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 15 / 30
Kirchhoff plate bending problem Equivalent variational formulation in ( 2 p Q = H0,γ 1 c γ s (), φ Ψ p H ()) 1, w Q = H 1 (). 0,γc γs 1 Find p Q such that p v dy = g v dx for all v Q. 2 For given p Q, find φ Ψ p such that C 1 symcurl φ : symcurl ψ dy = G[p], ψ for all ψ Ψ 0. 3 For given M = pi + symcurl φ, find w Q such that w q dy = H[M], q for all q Q. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 16 / 30
Decomposition of the problem 2 for φ = ( φ 2, φ 1 ) T we obtain Ĉ 1 ε(φ ) : ε(ψ) dy = Ĝ[p], ψ for all ψ Ψ 0 with an appropriately rotated material tensor Ĉ 1 and ε αβ (ψ) = 1 2 ( βψ α + α ψ β ) decomposition of the Kirchhoff model into 1 Poisson problem 2 plane linear elasticity problem 3 Poisson problem Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 17 / 30
Mindlin-Reissner plates Mindlin-Reissner model J = 1 { Cε(θ) : ε(θ) + µ t 2 w θ 2} dx 2 g w dx min Kirchhoff model: θ = w. J = 1 { } C 2 w : 2 w 2 dx g w dx min decomposition of the Mindlin-Reisser model into 1 Poisson problem 2 Stokes-like problem 3 Poisson problem Brezzi/Fortin 1986 Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 18 / 30
Kirchhoff-Love shells variational formulation of the linearized model: find u W such that ] [t Cε(u) : ε(v) + t3 a Cκ(u) : κ(v) dy = F, v for all v W 12 with the membrane strain the bending strain W H 1 () H 1 () H 2 (), ε αβ (u) = 1 2 (u α β + u β α ) b αβ u 3, κ αβ (u) = u 3 αβ b σ αb σβ u 3 + b σ αu σ β + b τ β u τ α + bτ β α u τ = αβ u 3 +.... Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 19 / 30
Kirchhoff-Love shells quantities of interest: u displacement M = t3 12 Cκ(u) a bending moments M = Ĉκ(u) Mixed variational formulation: Find M V and u Q such that Ĉ 1 M : L dy div Div L, u 3 L : κ 1 (u) dy = 0 div Div M, v 3 M : κ 1 (v) dy t Cɛ(u) : ɛ(v) a dy = F, v for all L V and v Q, with the function spaces where Q i H 1 (). V = H(div Div, ; Q 3 ) sym, Q = Q 1 Q 2 Q 3, Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 20 / 30
Numerical results benchmark examples of the shell obstacle course: Scordelis-Lo roof pinched cylinder pinched hemisphere midsurface represented as B-spline surfaces approximation spaces: equal-order approximation for u i, p, φ single patch: B-splines of maximum smoothness multipatch: C 0 implementation in G+Smo (C++ library for IgA) Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 21 / 30
Numerical results Example: Scordelis-Lo roof Figure: Geometry and load Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 22 / 30
u Numerical results Example: Scordelis-Lo roof 0.35 Scordelis-Lo roof 0.3 0.25 0.2 0.15 0.1 0.05 p = 2 p = 3 p = 4 p = 5 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 D.O.F. Figure: Results, 1 patch Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 23 / 30
u Numerical results Example: Scordelis-Lo roof 0.35 Scordelis-Lo roof (4 patches) 0.3 0.25 0.2 0.15 0.1 0.05 p = 2 p = 3 p = 4 p = 5 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 D.O.F. Figure: Results, 4 patches Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 24 / 30
Numerical results Example: pinched Cylinder Figure: Geometry and load Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 25 / 30
Numerical results Example: pinched Cylinder 2 Pinched cylinder (4 patches) #10-5 1.8 1.6 1.4 u 1.2 1 0.8 0.6 p=2 p=3 p=4 p=5 0.4 0.2 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 D.O.F. Figure: Results, 4 patches Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 26 / 30
Numerical results Example: pinched Hemisphere Figure: Geometry and load Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 27 / 30
u Numerical results Example: pinched Hemisphere 0.1 Pinched hemisphere (4 patches) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 p = 2 p = 3 p = 4 p = 5 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 D.O.F. Figure: Results, 4 patches Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 28 / 30
Conclusion and outlook Kirchhoff plate bending problems and similar 4-th order problems can be decomposed in three (consecutively to solve) second-order problems. extension to Kirchhoff-Love shell: formulation in H 1 spaces. work in progress: extend the numerical analysis from Kirchhoff plates to Kirchhoff-Love shells. behavior of equal-order discretization method w.r.t. membrane locking extension to Mindlin-Reissner plates and shells Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 29 / 30
References W. Krendl, K. Rafetseder, and W. Z. A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method. ETNA, Electron. Trans. Numer. Anal., 45:257 282, 2016. K. Rafetseder and W. Z. A decomposition result for Kirchhoff plate bending problems and a new discretization approach. arxiv:1703.07962. ArXiv e-prints, March 2017. Walter Zulehner (JKU Linz) Kirchhoff-Love plates and shells AANMPDE 10 30 / 30